\(\int \frac {(a+b x^3)^5}{x^{28}} \, dx\) [273]

Optimal result

Integrand size = 13, antiderivative size = 69 \[ \int \frac {\left (a+b x^3\right )^5}{x^{28}} \, dx=-\frac {a^5}{27 x^{27}}-\frac {5 a^4 b}{24 x^{24}}-\frac {10 a^3 b^2}{21 x^{21}}-\frac {5 a^2 b^3}{9 x^{18}}-\frac {a b^4}{3 x^{15}}-\frac {b^5}{12 x^{12}} \]

[Out]

-1/27*a^5/x^27-5/24*a^4*b/x^24-10/21*a^3*b^2/x^21-5/9*a^2*b^3/x^18-1/3*a*b^4/x^15-1/12*b^5/x^12

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {\left (a+b x^3\right )^5}{x^{28}} \, dx=-\frac {a^5}{27 x^{27}}-\frac {5 a^4 b}{24 x^{24}}-\frac {10 a^3 b^2}{21 x^{21}}-\frac {5 a^2 b^3}{9 x^{18}}-\frac {a b^4}{3 x^{15}}-\frac {b^5}{12 x^{12}} \]

[In]

Int[(a + b*x^3)^5/x^28,x]

[Out]

-1/27*a^5/x^27 - (5*a^4*b)/(24*x^24) - (10*a^3*b^2)/(21*x^21) - (5*a^2*b^3)/(9*x^18) - (a*b^4)/(3*x^15) - b^5/
(12*x^12)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^5}{x^{10}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {a^5}{x^{10}}+\frac {5 a^4 b}{x^9}+\frac {10 a^3 b^2}{x^8}+\frac {10 a^2 b^3}{x^7}+\frac {5 a b^4}{x^6}+\frac {b^5}{x^5}\right ) \, dx,x,x^3\right ) \\ & = -\frac {a^5}{27 x^{27}}-\frac {5 a^4 b}{24 x^{24}}-\frac {10 a^3 b^2}{21 x^{21}}-\frac {5 a^2 b^3}{9 x^{18}}-\frac {a b^4}{3 x^{15}}-\frac {b^5}{12 x^{12}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^{28}} \, dx=-\frac {a^5}{27 x^{27}}-\frac {5 a^4 b}{24 x^{24}}-\frac {10 a^3 b^2}{21 x^{21}}-\frac {5 a^2 b^3}{9 x^{18}}-\frac {a b^4}{3 x^{15}}-\frac {b^5}{12 x^{12}} \]

[In]

Integrate[(a + b*x^3)^5/x^28,x]

[Out]

-1/27*a^5/x^27 - (5*a^4*b)/(24*x^24) - (10*a^3*b^2)/(21*x^21) - (5*a^2*b^3)/(9*x^18) - (a*b^4)/(3*x^15) - b^5/
(12*x^12)

Maple [A] (verified)

Time = 3.62 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {a^{5}}{27 x^{27}}-\frac {5 a^{4} b}{24 x^{24}}-\frac {10 a^{3} b^{2}}{21 x^{21}}-\frac {5 a^{2} b^{3}}{9 x^{18}}-\frac {a \,b^{4}}{3 x^{15}}-\frac {b^{5}}{12 x^{12}}\)	\(58\)
norman	\(\frac {-\frac {1}{27} a^{5}-\frac {5}{24} a^{4} b \,x^{3}-\frac {10}{21} a^{3} b^{2} x^{6}-\frac {5}{9} a^{2} b^{3} x^{9}-\frac {1}{3} a \,b^{4} x^{12}-\frac {1}{12} b^{5} x^{15}}{x^{27}}\)	\(59\)
risch	\(\frac {-\frac {1}{27} a^{5}-\frac {5}{24} a^{4} b \,x^{3}-\frac {10}{21} a^{3} b^{2} x^{6}-\frac {5}{9} a^{2} b^{3} x^{9}-\frac {1}{3} a \,b^{4} x^{12}-\frac {1}{12} b^{5} x^{15}}{x^{27}}\)	\(59\)
gosper	\(-\frac {126 b^{5} x^{15}+504 a \,b^{4} x^{12}+840 a^{2} b^{3} x^{9}+720 a^{3} b^{2} x^{6}+315 a^{4} b \,x^{3}+56 a^{5}}{1512 x^{27}}\)	\(60\)
parallelrisch	\(\frac {-126 b^{5} x^{15}-504 a \,b^{4} x^{12}-840 a^{2} b^{3} x^{9}-720 a^{3} b^{2} x^{6}-315 a^{4} b \,x^{3}-56 a^{5}}{1512 x^{27}}\)	\(60\)

[In]

int((b*x^3+a)^5/x^28,x,method=_RETURNVERBOSE)

[Out]

-1/27*a^5/x^27-5/24*a^4*b/x^24-10/21*a^3*b^2/x^21-5/9*a^2*b^3/x^18-1/3*a*b^4/x^15-1/12*b^5/x^12

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{28}} \, dx=-\frac {126 \, b^{5} x^{15} + 504 \, a b^{4} x^{12} + 840 \, a^{2} b^{3} x^{9} + 720 \, a^{3} b^{2} x^{6} + 315 \, a^{4} b x^{3} + 56 \, a^{5}}{1512 \, x^{27}} \]

[In]

integrate((b*x^3+a)^5/x^28,x, algorithm="fricas")

[Out]

-1/1512*(126*b^5*x^15 + 504*a*b^4*x^12 + 840*a^2*b^3*x^9 + 720*a^3*b^2*x^6 + 315*a^4*b*x^3 + 56*a^5)/x^27

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^3\right )^5}{x^{28}} \, dx=\frac {- 56 a^{5} - 315 a^{4} b x^{3} - 720 a^{3} b^{2} x^{6} - 840 a^{2} b^{3} x^{9} - 504 a b^{4} x^{12} - 126 b^{5} x^{15}}{1512 x^{27}} \]

[In]

integrate((b*x**3+a)**5/x**28,x)

[Out]

(-56*a**5 - 315*a**4*b*x**3 - 720*a**3*b**2*x**6 - 840*a**2*b**3*x**9 - 504*a*b**4*x**12 - 126*b**5*x**15)/(15
12*x**27)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{28}} \, dx=-\frac {126 \, b^{5} x^{15} + 504 \, a b^{4} x^{12} + 840 \, a^{2} b^{3} x^{9} + 720 \, a^{3} b^{2} x^{6} + 315 \, a^{4} b x^{3} + 56 \, a^{5}}{1512 \, x^{27}} \]

[In]

integrate((b*x^3+a)^5/x^28,x, algorithm="maxima")

[Out]

-1/1512*(126*b^5*x^15 + 504*a*b^4*x^12 + 840*a^2*b^3*x^9 + 720*a^3*b^2*x^6 + 315*a^4*b*x^3 + 56*a^5)/x^27

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{28}} \, dx=-\frac {126 \, b^{5} x^{15} + 504 \, a b^{4} x^{12} + 840 \, a^{2} b^{3} x^{9} + 720 \, a^{3} b^{2} x^{6} + 315 \, a^{4} b x^{3} + 56 \, a^{5}}{1512 \, x^{27}} \]

[In]

integrate((b*x^3+a)^5/x^28,x, algorithm="giac")

[Out]

-1/1512*(126*b^5*x^15 + 504*a*b^4*x^12 + 840*a^2*b^3*x^9 + 720*a^3*b^2*x^6 + 315*a^4*b*x^3 + 56*a^5)/x^27

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{28}} \, dx=-\frac {\frac {a^5}{27}+\frac {5\,a^4\,b\,x^3}{24}+\frac {10\,a^3\,b^2\,x^6}{21}+\frac {5\,a^2\,b^3\,x^9}{9}+\frac {a\,b^4\,x^{12}}{3}+\frac {b^5\,x^{15}}{12}}{x^{27}} \]

[In]

int((a + b*x^3)^5/x^28,x)

[Out]

-(a^5/27 + (b^5*x^15)/12 + (5*a^4*b*x^3)/24 + (a*b^4*x^12)/3 + (10*a^3*b^2*x^6)/21 + (5*a^2*b^3*x^9)/9)/x^27